Question 160
Find all triples $(a, b, c)$ of positive integers such that
$$
\begin{aligned}
a^{2}+b^{2} &= n \operatorname{lcm}(a, b)+n^{2} \\
b^{2}+c^{2} &= n \operatorname{lcm}(b, c)+n^{2} \\
c^{2}+a^{2} &= n \operatorname{lcm}(c, a)+n^{2}
\end{aligned}
$$
for some positive integer $n$.
$$
\begin{aligned}
a^{2}+b^{2} &= n \operatorname{lcm}(a, b)+n^{2} \\
b^{2}+c^{2} &= n \operatorname{lcm}(b, c)+n^{2} \\
c^{2}+a^{2} &= n \operatorname{lcm}(c, a)+n^{2}
\end{aligned}
$$
for some positive integer $n$.